Gravitational Potential Energy



Gravitational Potential Energy

THEORY

Recall the change in potential energy of a system, (V, is related to the work, W, by the force one part of the system exerts on the other,

(U ( - W = Wext

where Wext is the work by an external force against the gravity force. We can think of this external work as the energy needed to stretch the gravitational bond between two masses.

Given two point masses, M and m, initially separated by a distance ri then M attracts m with a force of magnitude,

F = GMm/r2 .

If an external force is applied with the above magnitude it can move m from ri to a distance rf > ri . Then the external force does positive work on m given by,[pic]

rf

Wext = ( GMm dr/r2 = - GMm (1/rf - 1/ri ) > 0.

ri

Therefore, the change in gravitational potential energy of this two-particle system equals Wext this result,

(U = U(rr) – U(ri) ( Wext = - GMm (1/rf - 1/ri ).

Setting rf → ∞ and choosing the point of zero potential energy at infinity,

U(rf → ∞) ( 0 gives us on replacing ri → r,

U(r) = -GMm/r.

Note: r is in the denominator, not r2 and the reference location for potential energy is infinity, which then is the place of maximum potential energy.

If there is a collection of particles then there is an energy of this sort between each distinct “bond” between pairs of particles,

U(r) = ½ (i (j - G mi mj / rij ,

where in the second sum, j ( i, and the ½ is to avoid double counting of bonds.

APPLICATIONS OF THE ABOVE THEORY

( gravity inside earth r < R

To find the gravitational force on a point mass, m, at a distance r < R inside earth use Newton’s shell theorem (i.e. treat that portion of the earth at r < R as if it were concentrated at the center of the earth and ignore that portion at r > R). Note: one will need mass density of earth (ρ ( M / V) and the volume of a sphere (V ( 4πR3/3).

Mass (between r = 0 and r) = earth density x volume of sphere of radius r.

= (M/4πR3/3) x 4πr3/3 = M (r/R)3

( F(r ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download